research-article

Analysis of a queueing system with impatient customers and working vacations

Authors:

Gang XuAuthors Info & Claims

QTNA '11: Proceedings of the 6th International Conference on Queueing Theory and Network Applications

Pages 208 - 212

https://doi.org/10.1145/2021216.2021245

Published: 23 August 2011 Publication History

Abstract

In this paper, we consider an M/M/1 queueing system with working vacations and impatient customers, where we examine the case that customers' impatience is due to a working vacation. During the working vacation, customers are served at a slower than usual service rate and are likely to become impatient. Each customer, upon arrival, activates an independent "impatient timer", exponentially distributed. If a customer's service has not been completed before the customer's timer expires, the customer abandons the queue, never to return. We analyze this model and derive the probability generating functions of the number of customers present when the server is both in a service period and in a working vacation. We further obtain the closed-form expressions for various performance measures, including the mean system size, the mean sojourn time of a customer served, the proportion of customers served and the rate of abandonment due to impatience. Some numerical results are presented.

References

[1]

E. R. Obert. Reneging phenomenon of single channel queues. Mathematics of Operations Research, 4:162--178, 1979.

[2]

R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy and S. I. Ammar. Transient solution of the M/M/c queue with balking and reneging. Computers and Mathematics with Applications, 57:1280--1285, 2009.

Digital Library

[3]

T. Bonald and J. Roberts. Performance modeling of elastic traffic in overload. ACM Sigmetrics Performance Evaluation Review, 29:342--343, 2001.

Digital Library

[4]

N. Gans, G. Koole and A. Mandelbaum. Telephone call centers: Tutotial, review, and research prospects. Manufacturing and Service Operations Management, 5:79--141, 2003.

Digital Library

[5]

S. Benjaafar, J. Gayon and S. Tepe. Optimal control of a production-inventory system with customer impatience. Operations Research Letters, 38:267--272, 2010.

Digital Library

[6]

C. Palm. Methods of judging the annoyance caused by congestion. Tele, 4:189--208, 1953.

[7]

D. J. Daley. General customer impatience in the queue GI/G/1. Journal of Applied Aprobability, 2:186--205 1965.

[8]

L. Takacs. A single-server queue with limited virtual waiting time. Journal of Applied Probability 11:612--617, 1974.

[9]

F. Baccelli, P. Boyer and G. Hebuterne. Single-server queues with impatient customers. Advances in Applied Probability, 16:887--905, 1984.

[10]

O. J. Boxma and P. R. de Waal. Multiserver queues with impatient customers. ITC, 14:743--756, 1994.

[11]

B. Van Houdt, R. B. Lenin and C. Blonia. Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times. Queueing Systems, 45:59--73, 2003.

Digital Library

[12]

D. Yue and W. Yue. Analysis of M/M/c/N queueing system with balking, reneging and synchronous vacations. In: Wuyi Yue et al. (Eds.), Advanced in Queueing Theory and Network Applications. New York: Springer, 165--180, 2009.

[13]

E. Altman and U. Yechiali. Analysis of customers' impatience in queues with server vacations. Queueing Systems, 52:261--279, 2006.

Digital Library

[14]

E. Altman and U. Yechiali. Infinite-server queues with systems' additional task and impatient customers. Probability in the Engineering and Informational Sciences, 22:477--493, 2008.

Digital Library

[15]

U. Yechiali. Queues with system disasters and impatient customers when system is down. Queueing Systems, 56:195--202, 2007.

Digital Library

[16]

S. Economou and S. Kapodistria. Synchronized abandonments in a single server unreilable queue. European Journal of Operational Research 203:143--155, 2010.

[17]

N. Perel and U. Yechiali. Queues with slow servers and impatient customers. European Journal of Operational Research, 201: 247--258, 2010.

[18]

L. D. Servi and S. G. Finn. M/M/1 queues with working vacations (M/M/1/WV). Performance Evaluation, 50:41--52, 2002.

Digital Library

[19]

J. D. Kim, D. W. Choi and K. C. Chae. Analysis of queue-length distribution of the M/G/1 queue with working vacations. In: Hawaii International Conference on Statistics and Related Fields, 2003.

[20]

D. Wu and H. Takagi. M/G/1 queue with multiple working vacations, Performance Evaluation 63:654--681, 2006.

Digital Library

[21]

Y. Baba. Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters 33:201--209, 2005.

Digital Library

[22]

A. D. Banik, U. C. Gupta and S. S. Pathak. On the GI/M/1/N queue with multiple vacation-analytic analysis and computation. Applied Mathematical Modelling, 31:1701--1710, 2007.

[23]

M. Zhang and Z. Hou. Performance analysis of MAP/G/1 queue with working vacations and vacation interruption. Applied Mathematical Modelling 35:1551--1560, 2011.

Cited By

Yang DDai YWu C(2024)Equilibrium balking strategies in unobservable queues with multiple vacations and an optional serviceRAIRO - Operations Research10.1051/ro/202408358:3(2569-2589)Online publication date: 25-Jun-2024
https://doi.org/10.1051/ro/2024083
Choudhary AMahla NSharma D(2022)Cost Optimization of the Queueing System with Degrading Service Rate, Bernoulli Vacation, and a Regular Vacation After Fixed ServicesInternational Journal of Applied and Computational Mathematics10.1007/s40819-022-01319-z8:3Online publication date: 3-May-2022
https://doi.org/10.1007/s40819-022-01319-z
Sudhesh RAzhagappan A(2018)Transient analysis of an M/M/1 queue with variant impatient behavior and working vacationsOPSEARCH10.1007/s12597-018-0339-855:3-4(787-806)Online publication date: 20-Sep-2018
https://doi.org/10.1007/s12597-018-0339-8
Show More Cited By

Index Terms

Analysis of a queueing system with impatient customers and working vacations
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Numerical differentiation
  2. Probability and statistics
    1. Statistical paradigms
      1. Queueing theory
2. Theory of computation
  1. Design and analysis of algorithms

Recommendations

Queues with system disasters and impatient customers when system is down

Consider a system operating as an M/M/c queue, where c =1, 1< c < , or c = . The system as a whole suffers occasionally a disastrous breakdown, upon which all present customers (waiting and served) are cleared from the system and lost. A repair ...
Analysis of customers' impatience in queues with server vacations

Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In ...
A Multiserver Queueing System with Impatient Customers

Many real-world situations involve queueing systems in which customers wait for service for a limited time only and leave the system if service has not begun within that time. This paper considers a multiserver queueing system with impatient customers, ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Other conferences

QTNA '11: Proceedings of the 6th International Conference on Queueing Theory and Network Applications

August 2011

234 pages

ISBN:9781450307581

DOI:10.1145/2021216

Conference Chairs:
Bara Kim
Korea Univ., Korea
,
Ho Woo Lee
Sungkyunkwan Univ., Korea
,
Program Chairs:
Gang Uk Hwang
KAIST, Korea
,
Yang Woo Shin
Changwon National Univ., Korea

Copyright © 2011 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Korea University

In-Cooperation

SIGMETRICS: ACM Special Interest Group on Measurement and Evaluation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 August 2011

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Conference

QTNA '11

Sponsor:

QTNA '11: 6th International Conference on Queueing Theory and Network Applications

August 23 - 26, 2011

Seoul, Korea

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
322
Total Downloads

Downloads (Last 12 months)21
Downloads (Last 6 weeks)19

Reflects downloads up to 16 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Yang DDai YWu C(2024)Equilibrium balking strategies in unobservable queues with multiple vacations and an optional serviceRAIRO - Operations Research10.1051/ro/202408358:3(2569-2589)Online publication date: 25-Jun-2024
https://doi.org/10.1051/ro/2024083
Choudhary AMahla NSharma D(2022)Cost Optimization of the Queueing System with Degrading Service Rate, Bernoulli Vacation, and a Regular Vacation After Fixed ServicesInternational Journal of Applied and Computational Mathematics10.1007/s40819-022-01319-z8:3Online publication date: 3-May-2022
https://doi.org/10.1007/s40819-022-01319-z
Sudhesh RAzhagappan A(2018)Transient analysis of an M/M/1 queue with variant impatient behavior and working vacationsOPSEARCH10.1007/s12597-018-0339-855:3-4(787-806)Online publication date: 20-Sep-2018
https://doi.org/10.1007/s12597-018-0339-8
Azhagappan A(2018)Transient behavior of a Markovian queue with working vacation variant reneging and a waiting serverTOP10.1007/s11750-018-00495-wOnline publication date: 10-Dec-2018
https://doi.org/10.1007/s11750-018-00495-w
Sudhesh RAzhagappan ADharmaraja S(2017)Transient analysis of M/M/1 queue with working vacation, heterogeneous service and customers’ impatienceRAIRO - Operations Research10.1051/ro/201604651:3(591-606)Online publication date: 30-Jun-2017
https://doi.org/10.1051/ro/2016046
Selvaraju NGoswami C(2013)Impatient customers in an M/M/1 queue with single and multiple working vacationsComputers and Industrial Engineering10.1016/j.cie.2013.02.01665:2(207-215)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1016/j.cie.2013.02.016
Yu XWu DWu LLu YPeng J(2013)An M/M/1 Queue System with Single Working Vacation and Impatient CustomersThe 19th International Conference on Industrial Engineering and Engineering Management10.1007/978-3-642-38391-5_81(767-776)Online publication date: 14-Jun-2013
https://doi.org/10.1007/978-3-642-38391-5_81

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten